Triviality of the Peripheral Point Spectrum

If $T_t=\rme^{Zt}$ is a positive one-parameter contraction semigroup acting on $l^p(X)$ where $X$ is a countable set and $1\leq p <\infty$, then the peripheral point spectrum $P$ of $Z$ cannot contain any non-zero elements. The same holds for Feller semigroups acting on $L^p(X)$ if $X$ is locally compact.